In [1]:
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
import numpy as np
from IPython.display import display, Math, Latex
import pylab
import plot_methods
import optimization_methods
%matplotlib inline

Plot function graphic

In [2]:
my_function = lambda x, y: x**2+2*y**2+0.012*x*y-2*y+x
In [3]:
plot_methods.plot_function(my_function, 'My function')

Gradient descent

constant step

In [4]:
results_con_step = optimization_methods.gradient_descent_constant_step(my_function, [1,1], 1e-5, 0.001)
number of iteration: 2, current point: [0.99398205 0.99598408], function value: 1.9858626672233397
number of iteration: 3, current point: [0.99098213 0.99398822], function value: 1.9728967364860956
number of iteration: 4, current point: [0.98798824 0.99200038], function value: 1.9599987578859968
number of iteration: 5, current point: [0.98500036 0.99002052], function value: 1.9471683310148826
number of iteration: 6, current point: [0.98201848 0.98804862], function value: 1.9344050580935064
number of iteration: 7, current point: [0.97904259 0.98608464], function value: 1.9217085439530628
number of iteration: 8, current point: [0.97607267 0.98412855], function value: 1.9090783960163655
number of iteration: 9, current point: [0.97310871 0.98218032], function value: 1.896514224279188
number of iteration: 10, current point: [0.97015071 0.98023992], function value: 1.8840156412926525
number of iteration: 11, current point: [0.96719864 0.97830732], function value: 1.8715822621444627
number of iteration: 12, current point: [0.96425251 0.97638249], function value: 1.859213704441033
number of iteration: 13, current point: [0.96131229 0.97446539], function value: 1.8469095882896651
number of iteration: 14, current point: [0.95837797 0.97255599], function value: 1.8346695362807135
number of iteration: 15, current point: [0.95544954 0.97065426], function value: 1.822493173470089
number of iteration: 16, current point: [0.95252699 0.96876018], function value: 1.8103801273615332
number of iteration: 17, current point: [0.94961031 0.96687371], function value: 1.7983300278894752
number of iteration: 18, current point: [0.94669949 0.96499482], function value: 1.7863425074016113
number of iteration: 19, current point: [0.94379451 0.96312348], function value: 1.7744172006419623
number of iteration: 20, current point: [0.94089537 0.96125966], function value: 1.762553744733789
number of iteration: 21, current point: [0.93800204 0.95940333], function value: 1.7507517791628757
number of iteration: 22, current point: [0.93511452 0.95755446], function value: 1.7390109457606293
number of iteration: 23, current point: [0.9322328  0.95571302], function value: 1.727330888687562
number of iteration: 24, current point: [0.92935687 0.95387898], function value: 1.7157112544168367
number of iteration: 25, current point: [0.92648671 0.95205232], function value: 1.7041516917177884
number of iteration: 26, current point: [0.92362231 0.95023299], function value: 1.6926518516400129
number of iteration: 27, current point: [0.92076366 0.94842097], function value: 1.6812113874969614
number of iteration: 28, current point: [0.91791076 0.94661624], function value: 1.6698299548499507
number of iteration: 29, current point: [0.91506357 0.94481876], function value: 1.6585072114922714
number of iteration: 30, current point: [0.91222211 0.94302851], function value: 1.6472428174338387
number of iteration: 31, current point: [0.90938635 0.94124544], function value: 1.636036434884904
number of iteration: 32, current point: [0.90655628 0.93946955], function value: 1.62488772824124
number of iteration: 33, current point: [0.9037319  0.93770079], function value: 1.613796364068227
number of iteration: 34, current point: [0.90091318 0.93593915], function value: 1.602762011085626
number of iteration: 35, current point: [0.89810012 0.93418458], function value: 1.5917843401527003
number of iteration: 36, current point: [0.89529271 0.93243706], function value: 1.5808630242527038
number of iteration: 37, current point: [0.89249094 0.93069657], function value: 1.5699977384785364
number of iteration: 38, current point: [0.88969479 0.92896307], function value: 1.5591881600173707
number of iteration: 39, current point: [0.88690425 0.92723655], function value: 1.5484339681362207
number of iteration: 40, current point: [0.88411931 0.92551696], function value: 1.5377348441670855
number of iteration: 41, current point: [0.88133997 0.92380428], function value: 1.527090471492575
number of iteration: 42, current point: [0.8785662  0.92209849], function value: 1.5165005355316712
number of iteration: 43, current point: [0.87579801 0.92039955], function value: 1.5059647237251363
number of iteration: 44, current point: [0.87303536 0.91870744], function value: 1.4954827255216423
number of iteration: 45, current point: [0.87027827 0.91702214], function value: 1.485054232363313
number of iteration: 46, current point: [0.86752671 0.9153436 ], function value: 1.4746789376720075
number of iteration: 47, current point: [0.86478067 0.91367182], function value: 1.4643565368355569
number of iteration: 48, current point: [0.86204015 0.91200675], function value: 1.4540867271935451
number of iteration: 49, current point: [0.85930512 0.91034838], function value: 1.4438692080242364
number of iteration: 50, current point: [0.85657559 0.90869668], function value: 1.433703680530519
number of iteration: 51, current point: [0.85385153 0.90705161], function value: 1.4235898478265736
number of iteration: 52, current point: [0.85113294 0.90541316], function value: 1.4135274149246007
number of iteration: 53, current point: [0.84841981 0.90378129], function value: 1.403516088721665
number of iteration: 54, current point: [0.84571213 0.90215599], function value: 1.3935555779861515
number of iteration: 55, current point: [0.84300988 0.90053721], function value: 1.3836455933451446
number of iteration: 56, current point: [0.84031305 0.89892495], function value: 1.3737858472709994
number of iteration: 57, current point: [0.83762164 0.89731917], function value: 1.3639760540688175
number of iteration: 58, current point: [0.83493563 0.89571984], function value: 1.354215929863594
number of iteration: 59, current point: [0.83225501 0.89412694], function value: 1.344505192587207
number of iteration: 60, current point: [0.82957977 0.89254044], function value: 1.3348435619664243
number of iteration: 61, current point: [0.8269099  0.89096033], function value: 1.3252307595099082
number of iteration: 62, current point: [0.82424539 0.88938656], function value: 1.315666508495998
number of iteration: 63, current point: [0.82158622 0.88781913], function value: 1.3061505339604178
number of iteration: 64, current point: [0.8189324  0.88625799], function value: 1.2966825626838907
number of iteration: 65, current point: [0.8162839  0.88470313], function value: 1.2872623231801055
number of iteration: 66, current point: [0.81364071 0.88315452], function value: 1.2778895456838684
number of iteration: 67, current point: [0.81100283 0.88161214], function value: 1.26856396213888
number of iteration: 68, current point: [0.80837025 0.88007596], function value: 1.2592853061859657
number of iteration: 69, current point: [0.80574295 0.87854596], function value: 1.250053313151116
number of iteration: 70, current point: [0.80312092 0.8770221 ], function value: 1.2408677200340996
number of iteration: 71, current point: [0.80050415 0.87550438], function value: 1.231728265496777
number of iteration: 72, current point: [0.79789264 0.87399275], function value: 1.2226346898514264
number of iteration: 73, current point: [0.79528636 0.87248721], function value: 1.2135867350494043
number of iteration: 74, current point: [0.79268532 0.87098772], function value: 1.204584144669945
number of iteration: 75, current point: [0.7900895  0.86949425], function value: 1.1956266639084157
number of iteration: 76, current point: [0.78749889 0.8680068 ], function value: 1.1867140395657927
number of iteration: 77, current point: [0.78491347 0.86652532], function value: 1.1778460200369958
number of iteration: 78, current point: [0.78233325 0.8650498 ], function value: 1.1690223553003132
number of iteration: 79, current point: [0.7797582  0.86358021], function value: 1.1602427969059863
number of iteration: 80, current point: [0.77718832 0.86211653], function value: 1.1515070979658044
number of iteration: 81, current point: [0.7746236  0.86065874], function value: 1.142815013142163
number of iteration: 82, current point: [0.77206402 0.85920681], function value: 1.1341662986370586
number of iteration: 83, current point: [0.76950959 0.85776072], function value: 1.1255607121820623
number of iteration: 84, current point: [0.76696027 0.85632044], function value: 1.1169980130272532
number of iteration: 85, current point: [0.76441608 0.85488596], function value: 1.1084779619310887
number of iteration: 86, current point: [0.76187699 0.85345724], function value: 1.100000321149727
number of iteration: 87, current point: [0.75934299 0.85203427], function value: 1.0915648544270349
number of iteration: 88, current point: [0.75681408 0.85061702], function value: 1.0831713269840555
number of iteration: 89, current point: [0.75429024 0.84920547], function value: 1.0748195055092058
number of iteration: 90, current point: [0.75177147 0.8477996 ], function value: 1.066509158147761
number of iteration: 91, current point: [0.74925776 0.84639938], function value: 1.0582400544919937
number of iteration: 92, current point: [0.74674908 0.84500479], function value: 1.0500119655714892
number of iteration: 93, current point: [0.74424545 0.84361581], function value: 1.0418246638430693
number of iteration: 94, current point: [0.74174683 0.84223241], function value: 1.033677923180805
number of iteration: 95, current point: [0.73925323 0.84085458], function value: 1.0255715188666428
number of iteration: 96, current point: [0.73676464 0.83948229], function value: 1.0175052275806895
number of iteration: 97, current point: [0.73428103 0.83811552], function value: 1.0094788273913897
number of iteration: 98, current point: [0.73180241 0.83675425], function value: 1.001492097746257
number of iteration: 99, current point: [0.72932877 0.83539845], function value: 0.993544819462352
number of iteration: 100, current point: [0.72686008 0.8340481 ], function value: 0.9856367747170478
number of iteration: 101, current point: [0.72439636 0.83270319], function value: 0.9777677470383155
number of iteration: 102, current point: [0.72193757 0.83136368], function value: 0.9699375212960493
number of iteration: 103, current point: [0.71948372 0.83002957], function value: 0.9621458836924641
number of iteration: 104, current point: [0.71703479 0.82870081], function value: 0.9543926217531977
number of iteration: 105, current point: [0.71459078 0.82737741], function value: 0.9466775243182544
number of iteration: 106, current point: [0.71215167 0.82605932], function value: 0.9390003815329113
number of iteration: 107, current point: [0.70971745 0.82474654], function value: 0.9313609848391038
number of iteration: 108, current point: [0.70728812 0.82343904], function value: 0.9237591269662384
number of iteration: 109, current point: [0.70486366 0.82213679], function value: 0.9161946019226808
number of iteration: 110, current point: [0.70244407 0.82083979], function value: 0.9086672049869895
number of iteration: 111, current point: [0.70002933 0.819548  ], function value: 0.9011767326991775
number of iteration: 112, current point: [0.69761944 0.81826141], function value: 0.8937229828521389
number of iteration: 113, current point: [0.69521438 0.81697999], function value: 0.8863057544833299
number of iteration: 114, current point: [0.69281415 0.81570373], function value: 0.8789248478660376
number of iteration: 115, current point: [0.69041873 0.8144326 ], function value: 0.871580064501172
number of iteration: 116, current point: [0.68802812 0.81316658], function value: 0.864271207108851
number of iteration: 117, current point: [0.68564231 0.81190566], function value: 0.8569980796201737
number of iteration: 118, current point: [0.68326128 0.81064981], function value: 0.8497604871689435
number of iteration: 119, current point: [0.68088503 0.80939901], function value: 0.8425582360835941
number of iteration: 120, current point: [0.67851354 0.80815324], function value: 0.8353911338790221
number of iteration: 121, current point: [0.67614682 0.80691249], function value: 0.8282589892485206
number of iteration: 122, current point: [0.67378484 0.80567673], function value: 0.8211616120560092
number of iteration: 123, current point: [0.67142761 0.80444593], function value: 0.8140988133278781
number of iteration: 124, current point: [0.6690751  0.80322009], function value: 0.8070704052452502
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number of iteration: 126, current point: [0.66438423 0.80078319], function value: 0.7931160154675683
number of iteration: 127, current point: [0.66204585 0.79957208], function value: 0.7861896638383166
number of iteration: 128, current point: [0.65971216 0.79836585], function value: 0.7792969629707187
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number of iteration: 145, current point: [0.62074819 0.77858674], function value: 0.6670973123934212
number of iteration: 146, current point: [0.61849735 0.77746494], function value: 0.660780221656361
number of iteration: 147, current point: [0.61625102 0.77634766], function value: 0.6544935031720684
number of iteration: 148, current point: [0.6140092  0.77523487], function value: 0.6482369946182305
number of iteration: 149, current point: [0.61177188 0.77412657], function value: 0.6420105346480862
number of iteration: 150, current point: [0.60953905 0.77302272], function value: 0.6358139628841282
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number of iteration: 2761, current point: [-0.49704426  0.50148117], function value: -0.7529779759325809
number of iteration: 2762, current point: [-0.49705619  0.50148121], function value: -0.7529781180989021
number of iteration: 2763, current point: [-0.4970681   0.50148125], function value: -0.7529782596971434
number of iteration: 2764, current point: [-0.49707998  0.50148129], function value: -0.7529784007295743
number of iteration: 2765, current point: [-0.49709184  0.50148133], function value: -0.7529785411984561
number of iteration: 2766, current point: [-0.49710367  0.50148137], function value: -0.7529786811060403
number of iteration: 2767, current point: [-0.49711548  0.50148141], function value: -0.7529788204545702
number of iteration: 2768, current point: [-0.49712727  0.50148145], function value: -0.7529789592462794
number of iteration: 2769, current point: [-0.49713903  0.50148149], function value: -0.7529790974833932
number of iteration: 2770, current point: [-0.49715077  0.50148153], function value: -0.7529792351681275
number of iteration: 2771, current point: [-0.49716249  0.50148157], function value: -0.7529793723026896
number of iteration: 2772, current point: [-0.49717418  0.50148161], function value: -0.752979508889278
number of iteration: 2773, current point: [-0.49718585  0.50148165], function value: -0.7529796449300824
number of iteration: 2774, current point: [-0.49719749  0.50148169], function value: -0.7529797804272835
number of iteration: 2775, current point: [-0.49720912  0.50148173], function value: -0.7529799153830536
number of iteration: 2776, current point: [-0.49722072  0.50148177], function value: -0.7529800497995564
number of iteration: 2777, current point: [-0.49723229  0.50148181], function value: -0.7529801836789464
number of iteration: 2778, current point: [-0.49724385  0.50148185], function value: -0.7529803170233701
number of iteration: 2779, current point: [-0.49725538  0.50148189], function value: -0.7529804498349653
number of iteration: 2780, current point: [-0.49726688  0.50148193], function value: -0.7529805821158606
number of iteration: 2781, current point: [-0.49727837  0.50148197], function value: -0.7529807138681771
number of iteration: 2782, current point: [-0.49728983  0.501482  ], function value: -0.752980845094027
number of iteration: 2783, current point: [-0.49730127  0.50148204], function value: -0.7529809757955138
number of iteration: 2784, current point: [-0.49731268  0.50148208], function value: -0.7529811059747329
number of iteration: 2785, current point: [-0.49732407  0.50148212], function value: -0.7529812356337712
number of iteration: 2786, current point: [-0.49733544  0.50148216], function value: -0.7529813647747073
number of iteration: 2787, current point: [-0.49734679  0.5014822 ], function value: -0.7529814933996113
number of iteration: 2788, current point: [-0.49735811  0.50148224], function value: -0.7529816215105456
number of iteration: 2789, current point: [-0.49736942  0.50148228], function value: -0.7529817491095636
number of iteration: 2790, current point: [-0.4973807   0.50148232], function value: -0.752981876198711
number of iteration: 2791, current point: [-0.49739195  0.50148236], function value: -0.7529820027800254
number of iteration: 2792, current point: [-0.49740319  0.5014824 ], function value: -0.7529821288555357
number of iteration: 2793, current point: [-0.4974144   0.50148244], function value: -0.7529822544272634
number of iteration: 2794, current point: [-0.49742559  0.50148248], function value: -0.7529823794972214
number of iteration: 2795, current point: [-0.49743675  0.50148252], function value: -0.7529825040674145
number of iteration: 2796, current point: [-0.4974479   0.50148256], function value: -0.7529826281398401
number of iteration: 2797, current point: [-0.49745902  0.50148259], function value: -0.752982751716487
number of iteration: 2798, current point: [-0.49747012  0.50148263], function value: -0.7529828747993362
number of iteration: 2799, current point: [-0.4974812   0.50148267], function value: -0.7529829973903612
number of iteration: 2800, current point: [-0.49749225  0.50148271], function value: -0.752983119491527
number of iteration: 2801, current point: [-0.49750328  0.50148275], function value: -0.7529832411047913
number of iteration: 2802, current point: [-0.4975143   0.50148279], function value: -0.7529833622321035
number of iteration: 2803, current point: [-0.49752529  0.50148283], function value: -0.7529834828754054
number of iteration: 2804, current point: [-0.49753625  0.50148287], function value: -0.752983603036631
number of iteration: 2805, current point: [-0.4975472   0.50148291], function value: -0.752983722717707
number of iteration: 2806, current point: [-0.49755812  0.50148295], function value: -0.7529838419205518
number of iteration: 2807, current point: [-0.49756902  0.50148298], function value: -0.7529839606470761
number of iteration: 2808, current point: [-0.4975799   0.50148302], function value: -0.7529840788991838
number of iteration: 2809, current point: [-0.49759076  0.50148306], function value: -0.7529841966787699
number of iteration: 2810, current point: [-0.4976016  0.5014831], function value: -0.7529843139877234
number of iteration: 2811, current point: [-0.49761241  0.50148314], function value: -0.7529844308279241
number of iteration: 2812, current point: [-0.4976232   0.50148318], function value: -0.7529845472012454
number of iteration: 2813, current point: [-0.49763398  0.50148322], function value: -0.7529846631095531
number of iteration: 2814, current point: [-0.49764473  0.50148326], function value: -0.7529847785547052
number of iteration: 2815, current point: [-0.49765545  0.5014833 ], function value: -0.7529848935385525
number of iteration: 2816, current point: [-0.49766616  0.50148333], function value: -0.752985008062938
number of iteration: 2817, current point: [-0.49767685  0.50148337], function value: -0.7529851221296983
number of iteration: 2818, current point: [-0.49768751  0.50148341], function value: -0.7529852357406615
number of iteration: 2819, current point: [-0.49769815  0.50148345], function value: -0.7529853488976489
number of iteration: 2820, current point: [-0.49770877  0.50148349], function value: -0.7529854616024748
number of iteration: 2821, current point: [-0.49771938  0.50148353], function value: -0.7529855738569459
number of iteration: 2822, current point: [-0.49772995  0.50148357], function value: -0.7529856856628617
number of iteration: 2823, current point: [-0.49774051  0.5014836 ], function value: -0.7529857970220146
number of iteration: 2824, current point: [-0.49775105  0.50148364], function value: -0.7529859079361899
number of iteration: 2825, current point: [-0.49776156  0.50148368], function value: -0.7529860184071656
number of iteration: 2826, current point: [-0.49777206  0.50148372], function value: -0.7529861284367131
number of iteration: 2827, current point: [-0.49778253  0.50148376], function value: -0.7529862380265957
number of iteration: 2828, current point: [-0.49779299  0.5014838 ], function value: -0.7529863471785705
number of iteration: 2829, current point: [-0.49780342  0.50148383], function value: -0.7529864558943873
number of iteration: 2830, current point: [-0.49781383  0.50148387], function value: -0.752986564175789
number of iteration: 2831, current point: [-0.49782422  0.50148391], function value: -0.7529866720245116
number of iteration: 2832, current point: [-0.49783459  0.50148395], function value: -0.7529867794422839
number of iteration: 2833, current point: [-0.49784494  0.50148399], function value: -0.7529868864308276
number of iteration: 2834, current point: [-0.49785526  0.50148403], function value: -0.7529869929918583
number of iteration: 2835, current point: [-0.49786557  0.50148406], function value: -0.7529870991270844
number of iteration: 2836, current point: [-0.49787586  0.5014841 ], function value: -0.7529872048382069
number of iteration: 2837, current point: [-0.49788612  0.50148414], function value: -0.7529873101269209
number of iteration: 2838, current point: [-0.49789637  0.50148418], function value: -0.7529874149949141
number of iteration: 2839, current point: [-0.4979066   0.50148422], function value: -0.7529875194438677
number of iteration: 2840, current point: [-0.4979168   0.50148425], function value: -0.7529876234754561
number of iteration: 2841, current point: [-0.49792698  0.50148429], function value: -0.7529877270913472
number of iteration: 2842, current point: [-0.49793715  0.50148433], function value: -0.7529878302932018
number of iteration: 2843, current point: [-0.49794729  0.50148437], function value: -0.7529879330826746
number of iteration: 2844, current point: [-0.49795741  0.50148441], function value: -0.7529880354614134
number of iteration: 2845, current point: [-0.49796752  0.50148444], function value: -0.7529881374310591
number of iteration: 2846, current point: [-0.4979776   0.50148448], function value: -0.752988238993247
number of iteration: 2847, current point: [-0.49798766  0.50148452], function value: -0.7529883401496048
number of iteration: 2848, current point: [-0.49799771  0.50148456], function value: -0.7529884409017543
number of iteration: 2849, current point: [-0.49800773  0.50148459], function value: -0.7529885412513107
number of iteration: 2850, current point: [-0.49801773  0.50148463], function value: -0.7529886411998827
number of iteration: 2851, current point: [-0.49802771  0.50148467], function value: -0.7529887407490726
Precision: 9.982426344351942e-06

dynamic step

In [5]:
results_dyn_step = optimization_methods.gradient_descent(my_function, [1,1], 1e-5)
number of iteration: 2, current point: [-0.54205935 -0.02801692], function value: -0.19044503711757782
number of iteration: 3, current point: [-0.49881844  1.05669214], function value: -0.13651149009400831
number of iteration: 4, current point: [-0.50266956  0.48817176], function value: -0.752657727867009
number of iteration: 5, current point: [-0.50280241  0.50182808], function value: -0.753013307162182
number of iteration: 6, current point: [-0.50290919  0.50150074], function value: -0.7530135441436765
number of iteration: 7, current point: [-0.5030114  0.5015171], function value: -0.753013554108241
number of iteration: 8, current point: [-0.50300905  0.50150058], function value: -0.7530135541011796
number of iteration: 9, current point: [-0.50300902  0.50150923], function value: -0.7530135542438933
Precision: 8.652578261310483e-06

Contour

In [6]:
plot_methods.plot_contour(my_function, 'Contour')

Contour and Scatter

constant step contour

In [7]:
plot_methods.plot_contour_and_scatter_of_descent(my_function, results_con_step, 
                                                 'Contour and Scatter constant step')

dynamic step contour

In [8]:
plot_methods.plot_contour_and_scatter_of_descent(my_function, results_dyn_step, 
                                                 'Contour and Scatter dynamic step')
In [ ]: